Motion

• General view:

Deriving of motion equations in 3-Dimentional.

• Special view:

Some special example of motions.

Mechanics It is an important branch of physics and deals with the

effect of force on bodies.

It is further divided into two parts

1. Kinematics : it deals with the study of motion of bodies Without any reference to the cause of motion. 2. Dynamics : In dynamics we discuss the motion of bodies under the action of forces.

MOTION

1. what is motion?

when a body is continuously changing its position with respect to the surroundings (Frames of Reference) , then we say that the body is in motion.

EXAMPLE :

1.When an athlete is running on the ground then he is continuously changing his position with respect to the audience who are sitting at rest.

2.We are continuously changing our position since morning till night with respect to earth which is at rest.

3.The earth is continuously changing position with respect to sun which is at rest.

Frames of Reference

• The object or point from which movement is determined

• Movement is relative to an object that appears stationary

• Earth is the most common frame of reference

system of coordinatesUsually we show reference frame with coordinates

system. In three dimensions this is as:

x

y

z

1i

j

1

k 1

P

Ex: P = (-1.8, -1.4, 1.2) so the vector -1.8 i – 1.4 j + 1.2 k

will extend from the origin to P.

General view of motion, 3-D. Definitions

• Position ( p )

• Position Vector ( r

• Trajectory

• Distance (d• Displacement ( r )

• Position ( p ) – where you are located

start

stop

• Position Vector ( r )-An arrow from the center of coordinate to the position of body

x

zstart

stopri

rf y

start

stop

Trajectory – The path you drive (the path introduce from body movement)

Distance (d ) – how far you have traveled, regardless of direction ( the length of trajectory ) It is a scalar.

Displacement ( r ) –Chang in position of an object, ( An arrow from initial point to final point )

start

stop

r

ri

rf

r

x

y

z

start

stopri

rf

r

Distance vs. Displacement• You drive the path, and your odometer goes up

by 8 miles (your distance).• Your displacement is the shorter directed

distance from start to stop (green arrow).• What if you drove in a circle?

start

stop

SPEED

• Defined as –How fast you goAverage speed = distance travelled in a certain time

s = distance / time = d / t

• Distance usually measured in metres; time in seconds; speed in metres per second (m / s)

• Tells us nothing about the direction travelled in

• It is usually an “average” value (unless measured and recorded every second!)

speed

1

3

2

وابسته مقصد و مبدأ به متوسط تندی مقداراست

مکان زمان مسافت شده طی

متوسط تندی

1 t1 d11=0 ?

2 t2 d21 d21 /(t2- t1)

3 t3 d31 d31 /(t3- t1)

Speedهر • در اتومیبل سنج سرعت که است مقداری ای لحظه تندی

دهد می نشان لحظه

گرفت • نظر در متوسط تندی حد توان می را ای لحظه تندیخیلی مقصد و مبدأ طوریکه

باشند نزدیک هم به

S = Limdfi

tf - tif i

Velocity

• Velocity (v) – how fast and which way; the rate at which position changes

ذره • تغییربردارمکان نسبت متوسط سرعت ( مکان( تغییر این برای الزم زمان بر ذره بردارجابجایی

( است( = Vfiجابجایی∆r

tf - ti

=tf - ti

rf - ri

جابجایی • بردار تقسیم حاصل چون است برداری کمیت سرعت ( ) است جابجایی زمان عدد بر

Velocity

بردار • راستای با متوسط سرعت راستایاست یکی جابجایی

متوسط • سرعت راستای هم و مقدار هماست ومقصد مبدا به وابسته

1

43

r31 r412 r21

Velocityکه – است ای لحظه تندی همان ای لحظه سرعت

مشخصاست آن در حرکت راستایدر • متوسط سرعت حد توان می را ای لحظه سرعت

نزدیک هم به خیلی مقصد و مبدأ طوریکه گرفت نظرباشند

1

43

r31 r412 r21

باشد نزدیکتر مبدا به مقصد چه هر که است مشخص شکل درلذا شود می نزدیکتر مسیر بر مماس خط به جابجایی بردار

مماساست لحظه هر در حرکت مسیر بر ای لحظه سرعتنقاط مماسدر بردارهای شکل 3و 2در

Instantaneous velocity is always tangential to trajectory

V2

V3

V = Limtf - ti

rf - ri

f i

Velocityکه • شود می نتیجه ریاضیات در حد تعریف از

به نسبت مکان بردار مشتق ای لحظه سرعت. است زمان

V = dt

dr

مکان 1. بردار زمانی تابعیت دانستن با باال معادله r(t)از. آورد بدست را سرعت توان می

بدانیم • را سرعت زمانی تابعیت ای مساله در اگربدست را مکان زمانی تابعیت زیر صورت به میتوانیم

می آوریم انتگرالی مساله یک به تبدیل مساله چون البتهمثل ای درلحظه باید باشد r1(t1)مکان t1شود معین

dr = V × dt r(t) – r1(t1) = ∫V × dtt1

t

r(t) = ∫V × dt + r1(t1) t1

t

2. Slope of tangent of position or displacement time graph is

instantaneous velocity and its slope of chord is average velocity.

Speed vs. Velocity

• Average speed and the magnitude of average velocity are generally not equivalent because total distance and the absolute value of total displacement are generally not the same.

• The scalar absolute value (magnitude) of velocity is speed

• Average velocity magnitude is always smaller than or equal to average speed of a given particle

Example: Motion with constant velocity• Constant velocity means1. Direction of velocity is constant.2. Magnitude of velocity is constant.• Constant direction means: The motion is 1-D .• Constant magnitude means:

Then : X = V t + X0 ,

ثابت • سرعت با را مسیری فاصله نصف رفت موقع V1 متحرکی

ثابت سرعت با را باقیمانده نصف کند V2و می موقع طیسرعت با را برگشت زمان نصف با و V1برگشت را بقیه نصف

می V2سرعت کند. طیرفت؟ • موقع متوسط سرعتبرگشت؟ • موقع متوسط سرعتبرگشت؟ • و رفت در متوسط سرعت

V = V

Acceleration

• The rate of change of velocity is acceleration – how an object's speed or direction changes over time, and how it is changing at a particular point in time.

• Average acceleration: The average acceleration is the ratio between the change in velocity and the time interval.

Acceleration

• Instantaneous acceleration (the acceleration at an instant of time) is defined as the limiting value of average acceleration as Δt becomes smaller and smaller. Under such a limit, a → a.

• If we know velocity as a function of time V(t), we can calculate acceleration as a function of time.

Integral relations of motion

• The above definitions can be inverted by mathematical integration to find:

Some special view: Kinematics of constant acceleration

• When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly

accelerated motion. In this case, with t0=0

Additional relations in constant acceleration motion

V = V + V0

2

Drive these equations from equations of previous page

مساله

Example: One dimensional motion• If in start time of motion, the direction of initial velocity is the

same as the direction of acceleration, then we have a motion

in straight line. That is because

vf = v0 + a tSo the direction of Vf is the same as V0 and the direction of

motion doesn’t change.

If we show this direction by X then we have:

vf = v0 + a t vavg = (v0 + vf ) / 2

x = v0 t + ½ a t 2 vf2 – v0

2 = 2 a x

Acceleration due to Gravity

9.8 m/s2

Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance).

a = -g = -9.8 m/s2

This acceleration vector is the same on the way up, at the top, and on the way down!

ground

y

• vf = v0 - g t

• vavg = (v0 + vf ) / 2

y = - ½ g t 2 + v0 t

• vf2 – v0

2 = -2 g y

Velocity & Acceleration Sign ChartV E L O C I T Y

ACCELERATION

+ -

+ Moving forward;

Speeding up

Moving backward;

Slowing down

-

Moving forward;

Slowing down

Moving backward;

Speeding up

Sample Problems1. You’re riding a unicorn at 25 m/s and come to

a uniform stop at a red light 20 m away. What’s your acceleration?

2. A brick is dropped from 100 m up. Find its impact velocity and air time.

3. An arrow is shot straight up from a pit 12 m below ground at 38 m/s.

a. Find its max height above ground.

b. At what times is it at ground level?

Multi-step Problems

1. How fast should you throw a kumquat straight down from 40 m up so that its impact speed would be the same as a mango’s dropped from 60 m?

2. A dune buggy accelerates uniformly at

1.5 m/s2 from rest to 22 m/s. Then the brakes are applied and it stops 2.5 s later. Find the total distance traveled.

19.8 m/s

188.83 m

Answer:

Answer:

Graphing !x

t

A

B

C

A … Starts at home (origin) and goes forward slowly

B … Not moving (position remains constant as time progresses)

C … Turns around and goes in the other direction quickly, passing up home

1 – D Motion

Graphing w/ Acceleration

x

A … Start from rest south of home; increase speed gradually

B … Pass home; gradually slow to a stop (still moving north)

C … Turn around; gradually speed back up again heading south

D … Continue heading south; gradually slow to a stop near the starting point

t

A

B C

D

Tangent Lines

t

SLOPE VELOCITY

Positive Positive

Negative Negative

Zero Zero

SLOPE SPEED

Steep Fast

Gentle Slow

Flat Zero

x

On a position vs. time graph:

Increasing & Decreasing

t

x

Increasing

Decreasing

On a position vs. time graph:

Increasing means moving forward (positive direction).

Decreasing means moving backwards (negative direction).

Concavityt

x

On a position vs. time graph:

Concave up means positive acceleration.

Concave down means negative acceleration.

Special Points

t

x

PQ

R

Inflection Pt. P, R Change of concavity

Peak or Valley Q Turning point

Time Axis Intercept

P, STimes when you are at

“home”

S

Curve Summary

t

x

Concave Up Concave Down

Increasing v > 0 a > 0 (A)

v > 0 a < 0 (B)

Decreasing

v < 0 a > 0 (D)

v < 0 a < 0 (C)

A

BC

D

All 3 Graphs

t

x

v

t

a

t

Graphing Tips

• Line up the graphs vertically.

• Draw vertical dashed lines at special points except intercepts.

• Map the slopes of the position graph onto the velocity graph.

• A red peak or valley means a blue time intercept.

t

x

v

t

Graphing TipsThe same rules apply in making an acceleration graph from a velocity graph. Just graph the slopes! Note: a positive constant slope in blue means a positive constant green segment. The steeper the blue slope, the farther the green segment is from the time axis.

a

t

v

t

Area under a velocity graphv

t

“forward area”

“backward area”

Area above the time axis = forward (positive) displacement.

Area below the time axis = backward (negative) displacement.

Net area (above - below) = net displacement.

Total area (above + below) = total distance traveled.

Area

The areas above and below are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this too.

v

t

“forward area”

“backward area”

t

x

Area units

• Imagine approximating the area under the curve with very thin rectangles.

• Each has area of height width.• The height is in m/s; width is in

seconds.• Therefore, area is in meters!

v (m/s)

t (s)

12 m/s

0.5 s

12

• The rectangles under the time axis have negative

heights, corresponding to negative displacement.

Graphs of a ball thrown straight up

x

v

a

The ball is thrown from the ground, and it lands on a ledge.

The position graph is parabolic.

The ball peaks at the parabola’s vertex.

The v graph has a slope of -9.8 m/s2.

Map out the slopes!

There is more “positive area” than negative on the v graph.

t

t

t